MURNAGHAN-NAKAYAMA RULES FOR CHARACTERS OF IWAHORI-HECKE ALGEBRAS OF THE COMPLEX REFLECTION GROUPS G(r, p, n)

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1 Can. J. Math. Vol. 50 (), 998 pp MURNAGHAN-NAKAAMA RULES FOR CHARACTERS OF IWAHORI-HECKE ALGEBRAS OF THE COMPLE REFLECTION GROUPS G(r, p, n) TOM HALVERSON AND ARUN RAM ABSTRACT. Iwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike [AK], Broué and Malle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in oung s seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike [AK] and Ariki [Ari].. Introduction. The finite irreducible complex reflection groups come in three infinite families: the symmetric groups S n on n letters; the wreath product groups Z r o S n,wherez r denotes the cyclic group of order r; and a series of index-p subgroups G(r, p, n) ofz r os n for each positive integer p that divides r. In the classification of finite irreducible reflection groups, besides these infinite families S n, Z r,andg(r,p,n), there exist only 34 exceptional irreducible reflection groups, see [ST]. A formula for the irreducible characters of the Iwahori-Hecke algebras for S n is known [Ram], [KW], [vdj]. This formula is a q-analogue of the classical Murnaghan- Nakayama formula for computing the irreducible characters of S n. Similar formulas for the characters of the groups G(r, p, n) are classically known, see [Mac], [Ste], [AK], [Osi] and the references there. Formulas of this type are also known for the Iwahori-Hecke algebras of Weyl groups of types B and D [HR], [Pfe], [Pfe2]. Recently, Iwahori-Hecke algebras have been constructed for the groups Z r os n and G(r, p, n) [AK], [BM], [Ari]. In this paper we derive Murnaghan-Nakayama type formulas for computing the irreducible characters of the Iwahori-Hecke algebras that correspond to Z r o S n and G(r, p, n). Hoefsmit [Hfs] has given explicit analogues of oung s seminormal representations for the Iwahori-Hecke algebras of types A n, B n,andd n. Ariki and Koike, [AK] and [Ari], have constructed Hoefsmit-analogues of oung s seminormal representations Received by the editors December 5, 995. The second author gratefully acknowledges support from NSF grant DMS and from a research fellowship under Australian Research Council grant No. A AMS subject classification: Primary: 20C05; secondary: 05E05. c Canadian Mathematical Society

2 68 TOM HALVERSON AND ARUN RAM for Iwahori-Hecke algebras H r,p,n of the groups G(r, p, n). Our approach is to derive the Murnaghan-Nakayama rules by computing the sum of diagonal matrix elements in an explicit Hoefsmit representation of each algebra. We are motivated by Curtis Greene [Gre], who takes this approach using the oung seminormal form of the irreducible representations of the symmetric group and gives a new derivation of the classical Murnaghan-Nakayama rule. Greene does this by using the Möbius function of a poset that is determined by the partition which indexes the irreducible representation. We generalize Greene s poset theorem so that it works for our cases. In this way we are able to compute the characters of the Hecke algebras H r,n H r,,n. To compute the characters of the Iwahori-Hecke algebra H r,p,n of G(r, p, n), p Ù, we use double centralizer methods (Clifford theory methods) to write these characters in terms of a certain bitrace on the irreducible representations of H r,n H r,,n.wethen compute this bitrace in terms of the irreducible character values of H r,n. The character formulas given in this paper contain the Murnaghan-Nakayama rules for the complex reflection groups G(r, p, n) and the Iwahori-Hecke algebras of classical type as special cases. REMARK. In this paper we only give formulas for computing the characters of certain standard elements of the Iwahori-Hecke algebra which are given by (2.0) in the case of the Iwahori-Hecke algebras of Z r o S n and by (3.5) in the case of the Iwahori- Hecke algebras of G(r, p, n), p Ù. In this paper we have not made any effort to show that this is sufficient to determine the values of the characters on all elements. We have a method for proving this which will be given in another paper. Results of this type for Iwahori-Hecke algebras of Weyl groups have been given in [GP]. 2. Characters of Iwahori-Hecke Algebras of (ZÛrZ) o S n. For positive integers r and n,lets n denote the symmetric group of order n generated by s 2, s 3,...,s n,wheres i denotes the transposition s i (i, i), and let Z r ZÛrZ denote the finite cyclic group of order r. Then the wreath product group Z r o S n is a complex reflection group that can be identified with the group of all n ð n permutation matrices whose non-zero entries are r-th roots of unity. Let q and u, u 2,...,u r be indeterminates. Let H r,n be the associative algebra with over the field C(u, u 2,...,u r,q) given by generators T, T 2,...,T n and relations () T i T j T j T i, forji jjù, (2) T i T i+ T i T i+ T i T i+, for2 i n, (3) T T 2 T T 2 T 2 T T 2 T, (4) (T u )(T u 2 ) ÐÐÐ(T u r ) 0, (5) (T i q)(t i + q ) 0, for 2 i n. Upon setting q andu i ò i,whereòis a primitive r-th root of unity, one obtains the group algebra C[Z r o S n ]. The algebras H r,n were first constructed by Ariki and Koike [AK], and they were classified as cyclotomic Hecke algebras of type B n by Broué and Malle [BM]. In the special case where r andu, we have T, and

3 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 69 H,n is isomorphic to an Iwahori-Hecke algebra of type A n. The case H 2,n when r 2, u p,andu 2 p, is isomorphic to an Iwahori-Hecke algebra of type B n. Shapes and standard tableaux. As in [Mac], we identify a partition ã with its Ferrers diagram and say that a box b in ã is in position (i, j) inãif b is in row i and column j of ã. The rows and columns of ã are labeled in the same way as for matrices. An r-partition of size n is an r-tuple, ñ (ñ (), ñ (2),...,ñ (r) ) of partitions such that jñ () j + jñ (2) j + ÐÐÐ+jñ (r) j n.ifó (ó (), ó (2),...,ó (r) ) is another r-partition, we write óñif ó (i) ñ (i) for i r. In this case,we say that ñûó (ñ () Ûó (), ó (2) Ûñ (2),..., ñ (r) Ûó (r) )isanr-skew shape. We refer to r-skew shapes and r-partitions collectively as shapes. If ï is a shape of size n, astandard tableau L (L (), L (2),...,L (r) )ofshapeïis a filling of the Ferrers diagram of ï with the numbers,2,...,n such that the numbers are increasing left to right across the rows and increasing down the columns of each L (i). For any shape ï, letl(ï) denote the set of standard tableaux of shape ï and, for each standard tableau L,letL(k) denote the box containing k in L. Representations. Define the content of a box b of a (possibly skew) shape ï (ï (),...,ï (r) )by (2. ) ct(b) u k q 2(j i), if bis in position (i, j) inï (k). For each standard tableau L of size n, define the scalar (T i ) LL by (2. 2) (T i ) LL q q ) ctl(i), for 2 i n. Note that (T i ) LL depends only on the positions of the boxes containing i and i inl. Let ï (ï (),...,ï (r) ) be a (possibly skew) shape of size n, and for each standard tableau L 2 L(ï), let v L denote a vector indexed by L. LetV ï be the C(u,...,u r,q)- vector space spanned by fv L j L 2 L(ï)g, so that the vectors v L form a basis of V ï. Define an action of H r,n on V ï by defining (2. 3) T v L ct L() v L, T i v L (T i ) LL v L + q +(T i ) LL vsi L, 2 i n, where s i L is the same standard tableau as L except that the positions of i and i are switched in s i L.Ifs i Lis not standard, then we define v si L 0. The following theorem is due to oung [ou] for the symmetric group S n,tohoefsmit [Hfs] for H,n, and to Ariki and Koike [AK] for H r,n, r ½ 2. THEOREM 2.4 ([OU],[HFS],[AK]). The modules V ï,whereïrunsoverall r-partitions of size n, form a complete set of nonisomorphic irreducible modules for H r,n. Hoefsmit elements. Define elements t i 2 H r,n,forin,by (2. 5) t i T i T i ÐÐÐT 2 T T 2 ÐÐÐT i T i.

4 70 TOM HALVERSON AND ARUN RAM To our knowledge, these elements were discovered by Hoefsmit in the case of the Iwahori-Hecke algebras of type B n and were rediscovered by Ariki and Koike for Iwahori-Hecke algebras of Z r os n. For each standard tableau L of size n, define the scalar (t i ) LL by (2. 6) (t i ) LL ct L(i), for i n. The following proposition is due to Hoefsmit for r, 2 and to Ariki and Koike for r Ù 2. PROPOSITION 2.7 ([HFS], PROP ; [AK], PROP. 3.6). t i on a vector v L, where L is a standard tableau, is t i v L (t i ) LL v L. For i n, the action of Furthermore, these elements commute: PROPOSITION 2.8 ([AK], LEMMA 3.3). The subalgebra n,r of H r,n generated by t, t 2,...,t n is an abelian subalgebra, i.e., the t i commute. Standard elements. For k Ú nand 0 i r (2. 9) R (i) k (t k) i T k+ T k+2 ÐÐÐT, define and, for each k n,definer (i) kk (t k) i. We say that an S n -sequence of length m is a sequence (,..., m ) satisfying Ú 2 Ú ÐÐÐ Ú m n, and we say that a Z r -sequence of length m is a sequence (i,...,i m ) satisfying 0 i j r for each j. ForanS n -sequence (,..., m )andaz r -sequence į (i,...,i m ), define (2. 0) T R(i ), R (i 2) +, 2 ÐÐÐR (i m) m +, m 2 H r,n. For example, in H 4,0 we have the standard element T (0,2,3,) (3,4,8,0) R(0),3 R(2) 4,4 R(3) 5,8 R() 9,0 T 2T 3 (t 4 ) 2 (t 5 ) 3 T 6 T 7 T 8 t 9 T 0. For k Ú nand 0 i r, we define (2. ) (i) k (L) (t k) i LL (T k+) LL (T k+2 ) LL ÐÐÐ(T ) LL, and for k n,wedefine (i) kk (L) (t k) i LL.Since(T j) LL depends only on the positions of the boxes j and j inl, the scalar (i) k (L) depends only on the positions of the boxes containing k, k +,..., in L. Let (,..., m ) be an S n -sequence and (i,...,i m ) be a denote the coefficient PROPOSITION 2.2. Z r -sequence, and let L be a standard tableau of size n. Let T Lþ v vl of v L in T v L.Then T Lþ v vl (i ), (L) (i 2) +,l 2 (L) ÐÐÐ (i m) m +, m (L).

5 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 7 In particular, for given sequences and,thevaluet Lþ v vl depends only on the positions and the linear order of the boxes in L. PROOF. This follows from the definition of the action of H r,n on standard tableaux and the fact (2.3) that when T i acts on a standard tableau L it affects only the positions of L containing i and i. The result follows, since t j acts as a scalar (Prop. 2.7), and otherwise is a product (from right to left) of a decreasing sequence of generators T i. T Characters. If L is a standard tableau (of any shape, possibly of skew shape) with n boxes, define (2. 3) (i) (L) (i),n (L), and for any shape ï (possibly skew), define (2. 4) (i) (ï) L2L(ï) (i) (L). In making these definitions, the actual values in the boxes of L do not matter, only their positions and their order relative to one another are relevant. Thus, the definitions make sense when the standard tableaux have values that form a subset of f,2,...g(with the usual linear order). For an r-partition ï,letü ï H r,n denote the character of the irreducible H r,n -representation V ï determined in Theorem 2.4. The following theorem is our analogue of the Murnaghan- Nakayama rule. THEOREM 2.5. Let (,..., m ) be an S n -sequence, (i,...,i m ) be a Z r - sequence, and suppose that ï is an r-partition of size n. Then ü ï H r,n (T ) (i) (ñ () ) (i2) (ñ (2) Ûñ () ) ÐÐÐ (im) (ñ (m) Ûñ (m ) ), ; ñ (0) ñ () ÐÐÐñ (m) ï where the sum is over all sequences of shapes ; ñ (0) ñ () ÐÐÐ ñ (m) ïsuch that jñ (j) Ûñ (j ) j j j j. PROOF. By Proposition 2.2 the character ü ï H r,n is given by ü ï H r,n (T ) T Lþ v vl (i ), (L) (i 2) +, 2 (L) ÐÐÐ (i m) m +, m (L). L2L(ï) L2L(ï) The result follows by collecting terms according to the positions occupied by the various segments of the numbers f,2,..., g, f +,..., 2 g,..., f m +,..., m g. In view of Theorem 2.5 it is desirable to give an explicit formula for the value of (i) (ï). To do so requires some further notations: The shape ï is a border strip if it is connected and does not contain two boxes which are adjacent in the same northwest-tosoutheast diagonal. This is equivalent to saying that ï is connected and does not contain any 2 ð 2 block of boxes. The shape ï is a broken border strip if it does not contain any

6 72 TOM HALVERSON AND ARUN RAM 2ð2 block of boxes. Therefore, a broken border strip is a union of connected components, each of which is a border strip. Drawing Ferrers diagrams as in [Mac], we say that a sharp corner in a border strip is a box with no box above it and no box to its left. A dull corner in a border strip is a box that has a box to its left and a box above it but has no box directly northwest of it. The picture below shows a broken border strip with two connected components where each of the sharp corners has been marked with an s and each of the dull corners has been marked with a d. s s s d d s FIGURE 2.6 The following theorem is proved using Corollary 4.4 of Theorem 4.6. We have placed these results in Section 4, because they stand on their own as results on planar posets. THEOREM 2.7. Let ï be any shape (possibly skew) with n boxes. Let CC be the set of connected components of ï, and let cc jccj be the number of connected components of ï. (a) If ï is not a broken border strip, then (k) (ï) 0; (b) If ï is a broken border strip, then (0) (ï) (q q ) cc q c(bs) ( q ) r(bs), bs2cc and, for k r, (k) (ï) ( q + q ) ct(s) cc ct(d) d2dc jdcj ð ( ) t e t ct(dc) hk t cc ct(sc) t 0 ð bs2cc q c(bs) ( q ) r(bs), where SC and DC denote the set of sharp corners and dull corners in ï, respectively, and if bs is a border strip, then r(bs) is the number of rows in bs, and c(bs) is the number of

7 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 73 columns in bs. The content ct(b) of a box b is as given in (2.). The function e t ct(dc) is the elementary symmetric function in the variables fct(d), d 2 DCg, and the function h k t cc ct(sc) is the homogeneous symmetric function in the variables fct(s), s 2 SCg. PROOF. Recall from (2.2) that (T k ) LL q q ) ctl(k). It follows from the definitions of (k) (L) in (2.3) and (2.4) that we may apply Corollary 4.4 with x b ct(b) for all boxes b in ï. For two boxes a and b in ï that are adjacent in a diagonal, we have ct(a)ct(b) (q q ) q q 0. Thus, (k) (ï) 0ifïis not a broken border strip. Furthermore, 8 (q q ) > < ct(a)ct(b) : (q q ) q q, ifa and b are adjacent in a row, 2 (q q ) q q, ifa and b are adjacent in a column. 2 The result now follows from Corollary Characters of Iwahori-Hecke Algebras of G(r, p, n). In this section we define the complex reflection groups G(r, p, n) and their Iwahori- Hecke algebras H r,p,n. The groups G(r, p, n) are normal subgroups of indexp in the groups G(r,,n), and the groups G(r,,n) are isomorphic to the wreath products Z r o S n.the corresponding Hecke algebras H r,p,n are subalgebras of H r,n. We compute the irreducible characters of H r,p,n in terms of the irreducible characters of H r,n, which are computed in Section 2. The complex reflection groups G(r, p, n). Let r, p, d, andnbe positive integers such that pd r. The complex reflection group G(r, p, n)isthesetofnðnmatrices such that (a) The entries are either 0 or r-th roots of unity. (b) There is exactly one nonzero entry in each row and each column. (c) The d-th power of the product of the nonzero entries is. The order of G(r, p, n) isgivenbyjg(r,p,n)j dr n n!andg(r,p,n) is a normal subgroup of G(r,,n)ofindexp. Let ê e 2ôiÛr be a primitive r-th root of unity. Then G(r, p, n) is generated by the elements s 0 ê p E + n E ii, s êe 2 + ê E 2 + n E ii, s j iâ j,j i 2 i 3 E ii + E (j )j + E j(j ), 2 j n,

8 74 TOM HALVERSON AND ARUN RAM where E ij denotes the n ð n matrix with a in the i-th row and j-th column and with all other entries 0. EAMPLE 3.. The following are important special cases of G(r, p, n). () G(,, n) S n, the symmetric group. (2) G(r,,n) Z r os n. (3) G(2,, n) WB n the Weyl group of type B. (4) G(2, 2, n) WD n the Weyl group of type D. The Hecke algebras. Let e 2ôiÛp be a primitive p-th root of unity, and let q and x Ûp 0,...,x Ûp d be indeterminates. Then H r,n is the associative algebra with over the field C(x Ûp 0,...,x Ûp d,q) given by generators T,...,T n, and relations () T i T j T j T i,forji jjù, (2) T i T i+ T i T i+ T i T i+,for2in, (3) T T 2 T T 2 T 2 T T 2 T, (4) (T p x 0 )(T p x ) ÐÐÐ(T p x d ) 0, (5) (T i q)(t i + q ) 0, for 2 i n. This is the same as the definition of the algebra H r,n in Section 2 except that we are using x Ûp k,0kd, 0 p, in place of u,...,u r.leth r,p,n be the subalgebra of H r,n generated by the elements (3. 2) a 0 T p, a T T 2 T, and a i T i, 2 i n. Ariki ([Ari], Proposition.6) shows that H r,p,n is an analogue of the Iwahori-Hecke algebra for the groups G(r, p, n). The special case H 2,2,n is isomorphic to an Iwahori-Hecke algebra of type D n. Shapes and tableaux. As above, r, p, d, andnare positive integers such that pd r. Weorganizeeachr-partition ï ofsizeninto d groupsofppartitions each, sothatwecan write ï (ï (k, ) ), for 0 k d and0 p, P where each ï (k, ) is a partition and k, jï (k, ) j n. It is convenient to view the partitions ï (k,0),...,ï (k,p ) as all lying on a circle so that we have d necklaces of partitions, each necklace with p partitions on it. In order to specify this arrangement, we shall say that ï is a (d, p)-partition. As in Section 2, we let L(ï) denote the set of standard tableaux of shape ï, and, for each standard tableau L, letl(i) denote the box containing i in L. Action on standard tableaux. Let ï (ï (k, ) )bea(d,p)-partition of size n. Since H r,p,n is a subalgebra of H r,n, the irreducible H r,n -representations V ï are (not necessarily irreducible) representations of H r,p,n. However, we can easily describe the action of H r,p,n on V ï by restricting the action of H r,n. With the given specializations of the u i, the content of a box b of ï, see (2.), is ct(b) x Ûp k q 2(j i), if bis in position (i, j) inï (k, ).

9 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 75 As in Section 2, we define, for each standard tableau L of size n, the scalar (T i ) LL q q ) ctl(i), for 2 i n. From (2.3) and (3.2), it follows that the action of H r,p,n on V ï is given by p a 0 v L ct L() ct vl x k v L, if 2 L (k, ), L() (3.3) a v L (T 2 ) LL v L + L, q +(T 2 ) LL vs2 ct s 2 L() a i v L (T i ) LL v L + q +(T i ) LL vsi L. Recall that t i T i ÐÐÐT 2 T T 2 ÐÐÐT i for i n, and define elements S i 2 H r,p,n, i n,by S a 0 t p, S 2 a a 2 t t 2, S i a i a i ÐÐÐa 4 a 3 a a 2 a 3 a 4 ÐÐÐa i a i t t i, for3 i n. It follows from the action of the t i (Prop. 2.7) that the action of S i on V ï is also diagonal and is given by (3.4) S v L ct L() p vl S i v L ct L() ct L(i) vl. Furthermore, since the t i commute (Prop. 2.8), it follows that the S i commute. A ZÛpZ action on shapes. Let ï (ï (k, ) )bea(d,p)-partition. We define an operation õ that moves the partitions on each circle over one position. Given a box b in position (i, j) of the partition ï (k, ) then õ(b) is the same box b except moved to be in position (i, j) of ï (k, +),where + is taken modulo p.themapõis an operation of order p and acts uniformly on the shape ï (ï (k, ) ), on standard tableaux L (L (k, ) ) of shape ï, and on the basis vector v L of V ï : õ(ï) (ï (k, +) ), õ(l) (L (k, +) ), and õ(v L ) v õ(l). We use the notation õ in each case, since the operation is always clear from context. In the last case, extend linearly to get the vector space homomorphism õ: V ï! V õ(ï).if bis a box in a shape ï, then (3. 5) ct õ(b) ct(b). LEMMA 3.6. The map õ: V ï! V õ(ï) is a H r,p,n -module isomorphism, i.e., õ commutes with the action of H r,p,n. PROOF. Since(T i ) LL, see (2.2), depends only on the row and column of boxes i and i and not on the position of the tableaux, we have (T i ) õl,õl (T i ) LL,forallin

10 76 TOM HALVERSON AND ARUN RAM and all standard tableaux L.Sinceõ(s 2 L) s 2 õ(l), it follows that a i v õl õ(a i v L ). (Note that, because is a p-th root of unity, T does not commute with õ but that a 0 does.) The set of transformations fõ ã j 0 ãp g defines an action of the cyclic group ZÛpZ on the set of (d, p)-partitions and on the set of vector spaces V ï. Irreducible representations. Fix a (d, p)-partition ï of size n, andletk ï be the stabilizer of ï under the action of ZÛpZ. The group K ï is a cyclic group of order jk ï j and is generated by the transformation õ f ï where fï is the smallest integer between and p such that õ f ï (ï) ï. Thus, (3. 7) K ï fõ ãf ï :V ï!v ï j0ãjk ï j g. Figure (3.9) is an example of a (3, 6)-partition ï for which f ï 2andK ï f, õ 2, õ 4 g ¾ Z 3. The elements of K ï are all H r,p,n -module isomorphisms. The irreducible K ï -modules are all one-dimensional, and the characters of these modules are given explicitly by (3. 8) 7 ë j : K ï! C õ f ï! jf ï where 0 j jk ï j. To see this note that f ï is a primitive jkï j-th root of unity.,0 0,0 2,0 0,5 0,,5, 2,5 2,,2 0,4 0,3 0,2,4 2,4,3 FIGURE 3.9. A (3, 6)-partition ï with f ï 2. 2,3 2,2 It follows (from a standard double centralizer result) that as an H r,p,n ð K ï -bimodule M (3. 0) V ï jk ï j ¾ V (ï,j) Z j, j 0 where V (ï,j) is an H r,p,n -module and Z j is the irreducible K ï -module with character ë j. Ariki ([Ari], Theorem 2.6) has explicitly constructed the modules V (ï,j) and proved that they form a complete set of irreducible H r,p,n -modules. From the point of view of (3.0), one can prove that the V (ï,j) are irreducible H r,p,n -modules by setting q andx k for all 0 k d and appealing to the corresponding result for the group G(r, p, n).

11 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 77 THEOREM 3. ([ARI], THEOREM 2.6). The modules V (ï,j),whereïruns over all r- partitions and 0 j jk ï j, form a complete set of nonisomorphic irreducible modules for H r,p,n. REMARK 3.2. It should be noted that if f ï p and thus jk ï j, then the irreducible H r,n -module V ï is also an irreducible H r,p,n -module. Characters. Fix a (d, p)-partition ï and let ü (ï,j) denote the character of the irreducible H r,p,n -module V (ï,j) defined by (3.0). Let ü ï denote the H r,p,n ðk ï -bitrace on the module V ï, i.e.,ifh2h r,p,n and õ ãf ï 2 Kï,then (3. 3) ü ï (hõ ãf ï ) L2L(ï) hõ ãf ï v L þ vl L2L(ï) hv L þ võ ãf ï L, where hõ ãf ï vl þ vl denotes the coefficient of v L in the expansion of hõ ãf ï vl in terms of the basis of V ï corresponding to standard tableaux. By taking traces in the module equation (3.0), we obtain ü ï (hõ ãf ï ) jk ï j j 0 ü (ï,j) (h)ë j (õ ãf ï ) jk ï j j 0 ü (ï,j) (h) jãf ï. By the orthogonality of characters for K ï (or by direct computation) this formula can be inverted to give (3. 4) ü (ï,j) (h) jk ï j jãf ï ü ï (hõ ãf ï ), where f ï pûjk ï j. jk ï j ã 0 Standard elements. For k n, defines (i) kk Si k and define S (i) 2 Si a.forall other k Ú,define (3. 5) S (i) k Si k a k+ ÐÐÐa,and S (i) Si a a 3 ÐÐÐa. Following the definitions in (2.0), let (,..., m )beans n -sequence and let (i,...,i m ) be a Z r -sequence. The remainder of this section is devoted to computing the values ü (ï,j) ( S i S (i 2) +, 2 ÐÐÐS (i m) m +, m )andü (ï,j) (S i S (i 2) +, 2 ÐÐÐS (i m) m +, m ). ReductiontoR (i ), R (i 2) +, 2 ÐÐÐR (i m) m +, m. Recall the definition (2.9) of the element R (i) m, of H r,n. We now show that it is sufficient to compute characters on special products of these elements. LEMMA 3.6. The group generated by fa i j 0 i ng in H r,n is a normal subgroup of the group generated by ft j j j ng in H r,n. PROOF. It is sufficient to show that T j a i Tj and Tj a i T j can be written as a product of the a i s and their inverses. The only two nontrivial calculations are the following. T a 2 T T T 2 T T 2 T 2T T 2 T T 2 T T 2T T 2 a 2 a a 2

12 78 TOM HALVERSON AND ARUN RAM and T a T T 2 T 2 T 2 T (T T 2 T T 2 )T2 T T T 2T T 2 T T 2 T a a 2 a. LEMMA 3.7. For a word g Ti š ÐÐÐTi š m in the generators of H r,n (i.e., an element of the group in H r,n generated by the T i ), let å(g) denote the number of T s minus the number of T singsothatå(g)is the net number of T s in the word. Let h be in the group generated by fa i j 0 i ng in H r,p,n.then ü ï (ghg õ ãf ï ) ãf ïå(g) ü ï (hõ ãf ï ). PROOF. First note that, by Lemma 3.6, ghg 2 H r,p,n, so it makes sense to consider the bitrace. Then ü ï (ghg õ ãf ï ) ü ï (hg õ ãf ï g). We must bevery careful here, because,althoughthe action of h commutes with the action of õ ãf ï, the action of g and g do not. In fact, since () T v õl ct õl() v õl ct L() v õl õ(t v L ), and (2) T i v õl õ(t i v L ), for 2 i n, by Lemma 3.6, it follows that g õ ãf ï ãf ï å(g) õ ãf ï g. Thus, ü ï (ghg õ ãf ï ) ãf ïå(g) ü ï (ghõ ãf ï g ) ãf ïå(g) ü ï (hõ ãf ï ). LEMMA 3.8. Let (,..., m )be an S n -sequence, and let (i,...,i m )be a Z r -sequence, satisfying 0 i j r for each j. ü ï ( S i S (i 2) +, 2 ÐÐÐS (i m) m +, m õ ãf ï ) ãf ï(i 2 +ÐÐÐ+i m ) ü ï (R (i p, i 2 ÐÐÐ i m ) R (i 2) +, 2 ÐÐÐR (i m) m +, m õ ãf ï ), ü ï (S i S (i 2) +, 2 ÐÐÐS (i m) m +, m õ ãf ï ) ãf ï(i 2 +ÐÐÐ+i m ) ü ï (R (i p, i 2 ÐÐÐ i m ) R (i 2) +, 2 ÐÐÐR (i m) m +, m õ ãf ï ). PROOF. Wehave S (i) k t i R (i) k,ands(i) tip T 2 t T 3 ÐÐÐT. Since t commutes with T i for i Ù 2 it follows that, for any S n -sequence (,..., m )and Z r -sequence (i,...,i m ), we have S i S (i 2) +, 2 ÐÐÐS (i m) m +, m t i p T 2 t i 2 ÐÐÐ i m T 3 ÐÐÐT R (i 2) +, 2 ÐÐÐR (i m) m +, m S i S (i 2) +, 2 ÐÐÐS (i m) m +, m t i p T 2t i 2 ÐÐÐ i m T 3 ÐÐÐT R (i 2) +, 2 ÐÐÐR (i m) m +, m.

13 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 79 Both of these can be conjugated by a power of t to give Now use Lemma 3.7. R (i p i 2 ÐÐÐ i m ), R (i 2) +, 2 ÐÐÐR (i m) m +, m. In view of Lemma 3.8 and (3.4) we shall try to compute the values of ü ï (hõ ãf ï ), for elements h 2 H r,p,n of the form R (i ), R (i 2) +, 2 ÐÐÐR (i m) m +, m,wherei +ÐÐÐ+i m 0(modp), and where (,..., m )isans n -sequence and (i,...,i m )isaz r -sequence. In fact we shall prove the following theorem. We state the theorem now in order to establish the notations. THEOREM 3.9. Let ï be a (d, p)-partition, where pd r. Let ã be such that 0 ã jk ï j where K ï is as defined in (3.7). Define Let f ï pûjk ï j and ç jk ï j gcd(ã, jk ï j). h R (i ), R (i 2) +, 2 ÐÐÐR (i m) m +, m where (, ÐÐÐ, m )is an S n -sequence and (i, ÐÐÐ,i m )is a Z r -sequence such that i +ÐÐÐ+ i m 0 (mod p). The element h is an element of H r,p,n H r,n.ifall i in the sequence (,..., m )are divisible by ç then define n nûç, r rûç, p pûç, (,..., m ) ( Ûç,..., m Ûç), ï (k,ú) ï (k,ú), for 0 ú p and h R (0) ÐÐÐR (0)., m +, m Then: (a) If i is not divisible by ç for some i mthenü ï (hõ ãf ï ) 0. (b) If all i are divisible by ç and if i k  0 for some k, then ü ï (hõ ãf ï ) 0. (c) If all i are divisible by ç and if i k 0 for all k, then ü ï (hõ ãf ç n ï ) ü ï m H r, n 0 ( h) ç, [ç] n q i + q i A n where H r, n is with parameter q ç, in place of q and with parameters çú x ç k, 0 k d, 0 ú p gin place of u,...,u r. The element h is viewed as an element of the algebra H r, n and [ç] (q ç q ç )Û(q q ). REMARK The proofof this theoremwill occupythe remainderofthis section. Note that the case when ã 0, and thus ç, is particularly easy, since we have ü ï (hõ 0 ) ü ï H r,n (h), and these values are known by Theorem 2.7.

14 80 TOM HALVERSON AND ARUN RAM î-laced tableaux. Let the notations be as in Theorem 3.9, and let î ãf ï. Note that the orbit of a box in ï under the action of õ î is of size ç. Let w be the permutation given in cycle notation by (3. 2) w (,2,...,ç, ç)(ç +,ç+2,...,2ç)ððð. Define L(ï) î fl2ljõ î L w Lg. The elements of L(ï) î will be called î-laced tableaux of shape ï. It follows from (3.5) that if L is a î-laced tableau and j n then (3. 22) ct L(j) (mç j)î ct L(mç), where m is the positive integer such that 0 mç jç. As an example, the necklace in Figure 3.23 is part of a standard tableau that is 3-laced. (This is the analogue of the alternating tableaux defined in [HR].) Figure A necklace in a 3-laced standard tableau. LEMMA Let the notations be as in Theorem 3.9. If hv L þ võ î L Â 0, then (a) L is î-laced and (b) every i in the sequence (,..., m )is divisible by ç. PROOF. (a) Because of the special form of h the basis elements that appear in hv L are of the form v wl,wherew s j ÐÐÐs jk is a product of s j such that j Ú j 2 Ú ÐÐÐ Új k is a subset of the sequence f2,3,...,, +2, +3,..., 2, 2 +2,...g. This means that, in cycle notation, w is a product of cycles of the form (i, i +,i+ 2,...,j, j). Thus, hv L þ võ Â 0 only if õ î L wl for some permutation of this form. î L But any permutation ô such that ôl õ î L must have all cycles of length ç, it follows

15 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 8 that w w as given in (3.2). Thus, if hv L þ võ  0, then w î L õ î L and so L is L î-laced. (b) By the proof of (a), w s j ÐÐÐs jk where j Ú j 2 Ú ÐÐÐ Ú j k is a subset of the sequence of factors f2,3,...,, +2, +3,..., 2, 2 +2,...g. This fact and the explicit form of w in (3.2) implies that each i must be divisible by ç. Dividing by ç. Keeping the notations as in Theorem 3.9, let us now assume that the sequence (, ÐÐÐ, m ) is such that i is divisible by ç for all i. Let (3. 25) w j s i, i½j çâ j (i ) where the product is taken with the s i in increasing order and over all i ½ j such that i is not divisible by ç. Note that with this definition w is the same as given in (3.2) and that w L õ î L if hv L þ võ  0. In fact, it follows from the explicit form of h and the î L definition of the action in (3.3) that (3. 26) hv L þ võ F î j (L), L jn where F j (L) is defined as follows: (a) F j (L) (T j ) wj L,w j+ L,ifj is not divisible by ç, (b) F j (L) (T j ) wj+ L,w j+ L,ifj is divisible by ç but j  k for any k m, (c) F j (L) (t j ) i k wj+ L,w j+ L,ifj kfor some k m. We shall compute the values of the F j (L) explicitly in Lemma 3.30, but first we must introduce a bit more notation. Recall the definitions of n, r, p,(,..., m ), ï,and hin Theorem 3.9. If L is a î-laced tableau define integers ö,...,ö n and a (d, p) standard tableau L as follows: If L(mç) is in position (i, j) of the partition ï (k,ö m p+ú m ),then L(m) is in position (i, j) of the partition ï (k,úm). In the above ö m and ú m are chosen such that 0 ú m p. The map L 7! (ö,...,ö n, L) is a bijection between î-laced tableaux L and sequences (ö,...,ö n, L)where0ö m ç for each m nûç, and Lis a (d, p)-standard tableau. The following is the necklace of Figure 3.23 after dividing by ç (2,2,,2,2,0,0,0, ) Figure The 3-laced necklace of Figure 3.23 after division by 3.

16 82 TOM HALVERSON AND ARUN RAM For each j n, define d j ( öj mod ç, ifj k for some k; ö j ö j mod ç, otherwise; (an invertible linear transformation of (ZÛçZ) n ). Then the map L 7! (ö,...,ö n, L) 7! (d,...,d n, L), is a bijection between î-laced tableaux L and sequences (d,...,d n, L)where0d m ç for each m n,and Lis a (d, p)-standard tableau. The reason for introducing these bijections will become more clear in the proof of the following lemma. First let us define (3. 28) ct L(m) ú m x Ûp k q 2(j i), if m is in position (i, j) of the partition ï (k,úm) of L and then note that (3. 29) ct L(mç) ö m p ct L(m) ö m ct L(m), where p is a primitive ç-th root of unity. LEMMA Let the notations be as given in Theorem 3.9 and assume that î ãf ï, that L is a î-laced standard tableau, and that the sequence (,..., m ) is such that i is divisible by ç for all i. Let p e 2ôiÛç, and let F j (L) denote the factor defined in (3.26). Let j n and suppose that k is such that (k )ç Újkç. (a) If j is not divisible by ç,then q F j (L) (T j ) wj L,w j+ L (kç j+)î + q (kç j+)î. (b) If j is divisible by ç but j  i for any i m, F j (L) (T j ) wj+ L,w j+ L d ct k L(k ) ct L(k) q q (c) If j i i çfor some 0 i m, then F j (L) (t j ) i k wj+ L,w j+ L d i + ct L( i k. i +). PROOF. (a)ifj is not divisible by ç then w j+ L(j ) L(j ) and w j+ L(j) L(kç). Thus ct w j+ L(j ) ct kç L(j ) î (j )ct L(kç), and ct w j+ L(j) ct L(kç).

17 It follows that CHARACTERS OF IWAHORI-HECKE ALGEBRAS 83 F j (L) (T j ) wj L,w j+ L q + q q ctw j+ L(j ) ctw j+ L(j) q q q + î(kç j+) q (kç j+)î + q (kç j+)î. (b)if(k )ç j andj  i then w j+ L(j ) L (k )ç and w j+ L(j) L(kç), and ct w j+ L(j ) ct L (k )ç and ct w j+ L(j) ct L(kç). Thus F j (L) (T j ) wj+ L,w j+ L q q ö k ö k ct L(k ) ct L(k) q q ct L(k )ç ctl(kç) d ct k L(k ) ct L(k) q q (c) If j i i çthen w j+ L(j) L( i + ç) L ( i +)ç and ct w j+ L(j) ct L ( i +)ç Thus, F j (L) (t j ) i k wj+ L,w j+ L ct L ( i +)ç ik ö i + ct L( i +) i k d i + ct L( i +) i k. Note that the product of the factors of type (a) in the previous lemma satisfy (3. 3) C F j (L) ç, çâ j(j ) i q i + q A n i and thus we have that (3. 32) hv L þ võ F î j (L) C F k ( L), L jn k n wherewedefine F k ( L) F (k )ç+ (L). With this notation, the only factor in (3.32) which depends on the number d i is F i ( L)...

18 84 TOM HALVERSON AND ARUN RAM Proof of Theorem 3.9. PROOF. Letî ãf ï.then ü ï (hõ î ) hõ î v L þ vl L2L(ï) L2L( ï) d,...,d n 0 j C ç ç L2L(ï) F j (L) C n F k L2L( ï) d,...,d n 0 k n ç L2L( ï) k d k 0 F k ( L), hv õî Lþ vl hv L þ võ L2L(ï) î î L since the only factor in Q n k F k ( L) which depends on the number d i is F i ( L). (a) It follows from Lemma 3.24 that if there is some i that is not divisible by ç then ü ï (hõ î ) 0. (b) Suppose that all i are divisible by ç and that i k  0forsomekm.Let j k +.Then ç d k + 0 F k + ç d k + 0 ç d k + 0 and it follows that, if i k  0forsomek,then and ç (t j ) i k wj+ L,w j+ L ü ï (hõ î ) L2L î hv õî Lþ i kd k + ct L( k +) i k 0, vl 0. (c) Suppose that all i are divisible by ç and that all i k 0. Then d k + 0 ç d k 0 F k + ç d k + 0 (T j ) wj+ L,w j+ L ç (t j ) i k wj+ L,w j+ L ç d k 0 d k d k q q (q q )ç ç [ç] 0 ct L(k ) ct L(k) )ç ct L(k)ç B@ qç q ç ct L(m )ç ct L(m)ç + 0 ç CA

19 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 85 where j (k )ç and [ç] (q ç q ç )Û(q q ). It follows that if i k 0forallkthen ü ï (hõ î ) hv õî Lþ L2L(ï) î vl Cç m L2 L( ï) k n kâ i ç [ç] 0 B@ qç q ç ct L(k )ç ct L(k)ç With the definitions of H r, n as in the statement of the theorem, (2.) and Proposition (2.2) imply that Thus, L2 L( ï) k n kâ i 0 B@ qç q ç ct L(k )ç CA ü ï H r, n ( h). ct L(k)ç ü ï (hõ ãf ) C ç n [ç] n where H r, n is as in the statement of the theorem. m ü ï H r, n ( h), 4. The Poset Theorem. Curtis Greene [Gre] uses the theory of partially ordered sets (posets) and Möbius functions to prove a rational function identity ([Gre], Theorem 3.3) which can be used to derive the Murnaghan-Nakayama rule for symmetric group characters. In [HR], we modify Greene s theorem so that it can be applied to computing Murnaghan-Nakayama rules for the irreducible characters of the Iwahori-Hecke algebras of type A n, B n,and D n. In this section, we extend the poset theorem of [HR] so that it can be applied to computing Murnaghan-Nakayama rules (Theorem 2.7) for the irreducible characters of the cyclotomic Iwahori-Hecke algebras of type B. A poset is planar in the (strong) sense if its Hasse diagram may be order-embedded in RðRwithout edge crossings even when extra bottom and top elements are added (see [Gre] for details). A linear extension of a poset P is a poset L with the same underlying set as P and such that the relations in L form an extension of the relations in P to a total order. We will denote by L(P) the set of all linear extensions L of P. The Möbius function of a poset P is the function ñ: P ð P!Zdefined inductively for elements a, b 2 P by (4. ) ñ(a, b) ñ P (a,b) 8 > < >: (See [Sta] for more details on Möbius functions.) ifa b, PaxÚbñ(a,x) ifa Ú b, 0 ifa  b. CA.

20 86 TOM HALVERSON AND ARUN RAM Throughout this section ˆP will denote a planar poset with unique minimal element u and P ˆP fugwill be the poset obtained by removing the minimal element u from ˆP. We let SC be the set of minimal elements of P and we call these elements sharp corners. Two sharp corners s and s 2 of SC are adjacent if they are not separated by another sharp corner as the boundary of P is traversed. If s and s 2 are adjacent elements of SC and the least common multiple s _ s 2 exists, then we call s _ s 2 a dull corner of P.We let DC denote the set of all dull corners of P. Finally, we let cc denote the number of connected components of P, and note that cc jscj jdcj. Let fx a, a 2 ˆPg, be a set of commutative variables indexed by the elements of ˆP.For each 0 k r and each pair a Ú b in ˆP, define a weight, wt (k) (a, b), by wt (k) (a, b) x axb (4. 2) q q for all a, b 2 P, and wt (k) (u, a) xa k for all a 2 P. Then for any planar poset ˆP with unique minimal element u, define (4. 3) (k) (ˆP) wt (k) (a, b) ñˆp (a,b), a,b2ˆp aâ b where ñˆp (a, b)isthemöbius function for the poset ˆP. In [HR], Theorem 5.3, it is proved that (4. 4) (0) (ˆL) (0) (P)(q q ) cc, and (4. 5) ˆL2L( ˆP) ˆL2L( ˆP) () (ˆL) () (P)0 cc x s d2dc xd, The expansion in (4.5) is equal to zero if there is more than one connected component in P. The following is our extension of the poset theorem to include values of k Ù. THEOREM 4.6. Let ˆP be a planar poset (as defined above) with unique minimal element u. Let P ˆP nfug.then and, for k r, ˆL2L( ˆP) ˆL2L(ˆP) (0) (ˆL) (q q ) cc (0) (P), (k) (ˆL) (k) (P)( q + q ) cc x s jdcj ð t 0 d2dc xd ( ) t e t (x DC )h k t cc (x SC )

21 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 87 where cc is the number of connected components of P, e t (x DC ) is the elementary symmetric function in the variables fx d, d 2 DCg, and h k t cc (x SC ) is the homogeneous symmetric function in the variables fx s, s 2 SCg. PROOF. For each s 2 SC define ˆP s to be the same poset as ˆP except with the additional relations s s 0,fors s 0 2 SC, and all other relations implied by transitivity. Each poset P s is planar, and each linear extension of P s must place the sharp corner s (a minimal element of P) immediately after u in the ordering, so we have (k) (ˆL) (k) (ˆL s ) wt (k) (u, s) (k) (L s ), ˆL2L( ˆP) ˆL s 2L(ˆP s ) L s 2L(P s ) In P we have wt (k) (a, b) wt (0) (a, b), so by [HR], Theorem 5.3, the second sum can be computed as (k) (L s ) (0) (L s ) (0) (k) (P s ), L s 2L(P s ) L s 2L(P s ) since P s is connected. Moreover, wt (0) (u, s) for each s 2 SC, so the case k 0is proved. From now on assume that k r. Then we have (k) (ˆL) wt (k) (u, s) (k) (P s ) ˆL2L( ˆP) (k) (P) wt (k) (u, s) (k) (P s ) (k) (P) (4. 7) (k) (P) wt (k) (u, s) wt (k) (a, b) ñ Ps (a,b) a,b2p wt (k) (a, b) ñ P(a,b) aâ b (k) (P) wt (k) (u, s) wt (k) (a, b) ñ Ps (a,b) ñp(a,b). where ñ Pi (a, b)andñ P (a,b)arethemöbius functions for their respective posets. We use the work of Greene [Gre] to compute the differences ñ Ps (a, b) ñ P (a,b)for aúb2p.letp Ł and P Ł s denote the dual of P and P s, respectively (that is u P Ł v, v P u). Then, ñ P (u, v) ñ P Ł(v,u) (see [Sta], p. 20), so we want to compute a,b2p aâ b ñ P Ł s (b, a) ñ P Ł(b,a) for a Ú b 2 P. Using the Möbius notation of [Gre] (p. 8, formulas (7) and (8)), let é a ñ P Ł(t, a)t, and é (s) a ñ P Ł s (t,a)t, t P Ł a t P Ła s so that a é t, and a é (s) t. t P Ł a t P Ła s

22 88 TOM HALVERSON AND ARUN RAM In this way, é (s) a é a for all a 2 P Ł n SC, and é (s) s é s s 0 + s 0 2SCnfsg d2dc It follows that ñ P Ł s (s 0, s) ñ P Ł(s 0,s), for all s 0 2 SC nfsg, ñ P Ł s (d,s) ñ P Ł(d,s) +, for all d 2 DC, ñ P Ł s (a, b) ñ P Ł(a,b) 0, for all other a, b 2 P, and wt (k) (a, b) ñ Ps (a,b) ñp(a,b) wt (k) (s, s 0 ) wt (k) (s, d). a,b2p s 0 2SCnfsg d2dc aâ b Substituting back into (4.7) gives (k) (ˆL) (k) (P) wt (k) (u, s) wt (k) (s, s 0 ) wt (k) (s, d). ˆL2L( ˆP) s 0 2SCnfsg d2dc Using the fact that jscj jdcj cc (the number of connected components of p), we cancel factors of q q and factor out x s and xd as follows (k) (ˆL) (k) (P) xs k q q x s x d ˆL2L( ˆP) s 0 2SCnfsg x s xs 0 d2dc q q (P) (k) x s xd (q q ) cc d2dc (x d x s ) x k s d2dc (x s 0 x s ). s 0 2SCnfsg For notational convenience, let F (P) (k) x s xd (q q ) cc. d2dc Order the sharp corners s, s 2,...,s jscj from left to right as the boundary of P is traversed, and let js i j i(its position in the ordering). Then (x d x s ) (k) (ˆL) F x k s d2dc ˆL2L( ˆP) (x s 0 x s ) s 0 2SCnfsg F x k s ( ) jscj jsj (x d x s ) (x p x q ) d2dc púq2scnfsg (x s 0 x s ) (x s x s 0) (x p x q ) s 0 Ús s 0 Ùs púq2scnfsg (x d x s ) (x p x q ) F x k s ( ) jscj jsj d2dc púq2scnfsg, V(x SC ) d.

23 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 89 where V(x SC ) is the Vandermonde determinant in the variables fx s, s 2 SCg. Moreover, d2dc (x d x s ) jdcj t 0 e t (x DC )( ) jdcj t x jdcj t s, where e t (x DC ) is the elementary symmetric function in the variables fx d, d 2 DCg.Again for notational convenience, let We then have ˆL2L( ˆP) (k) (ˆL) F G (x p x q ). púq2scnfsg x k s ( ) jdcj F ( ) t e t (x DC ) F( t 0 ) jscj jsjjdcj jscj jdcj jdcj t 0 t 0 e t (x DC )( V(x SC ) ) jdcj t x jdcj s t G x jdcj t+k s ( ) jdcj+jscj jsj G ( ) t e t (x DC ) V(x SC ) xjdcj t+k s ( ) jsj G. V(x SC ) jdcj t+k k t (jscj jdcj)+jscj k t cc +jscj Notice that xs xs xs and that the numerator k t cc (4. 8) x +jscj s ( ) jsj (x p x q ) púq2scnfsg is the alternating symmetrization of the monomial k t cc x +jscj s x jscj 2 s 2 x jscj s 3 3 ÐÐÐx jscj jscj s jscj. When we divide the numerator (4.8) by the Vandermonde V(x SC ), we get the Schur function s (k t cc,0,0,...,0) (x SC ) or, equivalently, the homogeneous symmetric function h k t cc (x SC ), and so (k) cc (ˆL) F( ) jdcj ( ) t e t (x DC )h k t cc (x SC ), ˆL2L( ˆP) t 0 and the proof is completed. Special cases. The homogeneous symmetric function satisfies h m (x SC ) 0 unless m Ù 0, so if k ½, then h k t cc (x SC ) 0, unless cc k. In particular, when k, the poset P must be connected (cc ), and (4. 9) jdcj t 0 ( ) t e t (x DC )h k t cc (x SC ) e 0 (x DC )h 0 (x SC )

24 90 TOM HALVERSON AND ARUN RAM and (4. 0) ˆL2L( ˆP) () (ˆL) () (P) x s d2dc xd, which agrees with (4.5). Shapes and standard tableaux. Theorem 4.6 reduces the problem of computing PˆL2L( ˆP) (k) (ˆL) to computing (k) (P). In the case where P is the poset of a (skew) shape, the product (k) (P) is readily computed and has been done so for q by Greene [Gre] and for generic q in [HR]. The result uses the natural extension of the theory of shapes and tableaux to the theory of partially ordered sets. (For a full treatment of this subject, see [Sta], whose notation we use here). If ï is a shape (possibly skew), then we construct a corresponding poset P ï whose Hasse diagram is given by placing a node in each box of ï and then drawing edges connecting nodes in adjacent boxes. The order relation in this poset is so that the smallest nodes are in the upper left corners. For example, $ and $ Figure 4. Note that posets corresponding to shapes are always planar and that the sharp and dull corners that we defined for partitions and shapes (see Figure 2.9) are exactly the sharp and dull corners of the corresponding poset. THEOREM 4.2. ([Gre], Theorem 3.3; [HR], Theorem 5.8) Let P ï be the poset of any shape (or skew shape) ï,letfx b gbe a set of commutative variables indexed by fb 2 P ï g, and let q be an indeterminate. Define (4. 3) wt(a, b) x axb q q for all a, b 2 P ï and (P ï ) wt(a, b) ñ P (a,b) ï. Then (P ï ) D a,b2p ï aâ b x b x a q q! R! q q x b xa C! q q x b xa where D is the set of pairs (a, b) of boxes in ï adjacent (northwest to southeast) in a diagonal, R is the set of pairs (a, b) of boxes in ï adjacent (west to east) in a row, and

25 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 9 C is the set of pairs (a, b) of boxes in ï adjacent (north to south) in a column. Let ï be a shape (or a skew shape) and let L(ï) be the set of standard tableaux of shape ï. Linear extensions of the poset P ï are in one-to-one correspondence with standard tableauxhaving skewshape ï as follows: Given a standard tableau T of shapeï let T(k) denote the box containing k in T. Then the standard tableau T corresponds to the linear extension L of the poset P ï which has underlying set P ï and order relations given by T(k) L T(l)ifkl. We can identify the standard tableau T with the chain L. Let ˆP ï be the poset P ï [fugwhere the adjoined element u satisfies u a for all a 2 P ï. The linear extensions of the poset ˆP ï are in one-to-one correspondence with the linear extensions of the poset P ï. Thus, we can identify a standard tableau T of shape ï with a linear extension ˆL of the poset ˆP ï. Let ñ be the Möbius function of the linear extension ˆL of ˆP ï that corresponds to the standard tableau T of shape ï. Then, since ˆL is a chain, ñ satisfies It follows that ñ(a, b) (, if a Ú b and a is adjacent to b in ˆL, and 0, if a Ú b and a is not adjacent to b in ˆL. (k) (T) (k) (ˆL) aúb2ˆl wt (k) (a, b) ñ(a,b) (x T() ) k n i 2 (q q ) x T(i ) xt(i). COROLLAR 4.4. Let ï be any shape (or skew shape) with n boxes. Let fx b g be a set of commutative variables indexed by the boxes b 2ï, and let q be an indeterminate. Then (0) (T) (q q ) cc x b! xa! q q! q q T2L(ï) D q q R x b xa C x b xa, and, for k r, (k) (T) ( q+q ) cc x s xd T2L(ï) d2dc 0 B@ D jdcj ð t 0 ð ( ) t e t (x DC )h k t cc (x SC ) x b x a q q C A 0 B@ R q q x b xa CA 0 B@ C q q x b xa where cc is the number of connected components of ï, SC is the set of sharp corners of ï, DC is the set of dull corners of ï, R is the set of pairs (a, b) of boxes in ï adjacent (west to east) in a row, C is the set of pairs (a, b) of boxes in ï adjacent (north to south) in a column, and CA,

26 92 TOM HALVERSON AND ARUN RAM D is the set of pairs (a, b) of boxes in ï adjacent (northwest to southeast) in a diagonal. PROOF. LetP ï be the poset of the shape ï,andletˆp ï be the poset P ï [fug,wherethe adjoined element u satisfies u a for all a 2 P ï.thenˆp P ï is a planar poset with unique minimal element, so we apply Theorem 4.6 to compute T2L(ï) (k) (T) PˆL2L( ˆP ï ) (k) (ˆL). This reduces the problem to computing (k) (P ï ). Inside of P ï, the weights (4.2) are all independent of k and of the form (4.3), so we use Theorem 4.2 to compute (k) (P ï ). REFERENCES [Ari] S. Ariki, Representation theory of a Hecke algebra of G(r, p, n). J. Algebra 77(995), [AK] S. Ariki and K. Koike, A Hecke algebra of (ZÛrZ)oS n and construction of its irreducible representations. Adv. Math. 06(994), [BM] M. Broué and G. Malle, Zyklotomische Heckealgebren.Astérisque 22(993), [GP] M. Geck and G. Pfeiffer,On the irreducible characters of Hecke algebras. Adv. Math. 02(993), [Gre] C. Greene, A rational function identity related to the Murnaghan-Nakayama formula for the characters of S n.j.alg.comb.(992), [HR] T. Halverson and A. Ram, Murnaghan-Nakayama rules for the characters of Iwahori-Heckealgebras of classical type. Trans. Amer. Math. Soc. 348(996), [Hfs] P. N. Hoefsmit, Representations of Heckealgebras of finite groups with BN-pairs of classical type. Thesis, University of British Columbia, 974. [KW] R. C. King and B. G. Wybourne, Representations and traces of Hecke algebras H n (q) of type A n.j. Math. Phys. 33(992), 4 4. [Mac] I. G. Macdonald, Symmetric Functions and Hall Polynomials. Oxford University Press, 979. [Osi] M. Osima, On the representations of the generalized symmetric group. Math. J.Okayama Univ. 4(954), [Pfe] G. Pfeiffer,Character values of Iwahori-Heckealgebras of typeb. In: Finite Reductive Groups: Related Structures and Representations (Ed. M. Cabanes), Progress in Math. 4(996), [Pfe2], Charakterwerte von Iwahori-Hecke-Algebren von klassischem Typ. Aachener Beiträge zur Mathematik 4(995), Verlag der Augustinus-Buchhandlung, Aachen. [Ram] A. Ram, A Frobenius formula for the characters of the Hecke algebras. Invent. Math. 06(99), [ST] G. C. Shephard and J. A. Todd, Finite unitary reflection groups. Canad. J. Math. 6(954), [Sta] R. Stanley, Enumerative Combinatorics I. Wadsworth & Brooks/Cole, 986. [Ste] J. Stembridge, On the eigenvalues of representations of reflection groups and wreath products. Pacific J. Math. 40(989), [vdj] J. van der Jeugt, An algorithm for characters of Hecke algebras H n (q) of type A n.j.phys.a24(99), [ou] A. oung, Quantitative substitutional analysis I I. Proc. London Math. Soc Department of Mathematics Macalester College St. Paul, MN USA Department of Mathematics Princeton University Princeton, NJ USA